Physics 235
Chapter 11
Chapter 11
Dynamics of Rigid Bodies
A rigid body is a collection of particles with fixed relative positions, independent of the
motion carried out by the body. The dynamics of a rigid body has been discussed in our
introductory courses, and the techniques discussed in these courses allow us to solve many
problems in which the motion can be reduced to two-dimensional motion. In this special case,
we found that the angular momentum associated with the rotation of the rigid object is directed
in the same direction as the angular velocity:
L = I!
In this equation, I is the moment of inertia of the rigid body which was defined as
I = ! mi ri 2
i
where ri is the distance of mass mi from the rotation axis. We also found that the kinetic energy
of the body, associated with its rotation, is equal to
T=
1 2
I!
2
The complexity of the motion increases when we need three dimensions to describe the motion.
There are many different ways to describe motion in three dimensions. One common method is
to describe the motion of the center of mass (in a fixed coordinate system) and to describe the
motion of the components around the center of mass (in the rotating coordinate system).
The Inertia Tensor
In Chapter 10 we derived the following relation between the velocity of a particle in the fixed
reference frame, vf, and its velocity in the rotating reference frame vr:
v f = V + vr + ! " r
If we assume that the rotating frame is fixed to the rigid body, then vr = 0.
The total kinetic energy of the rigid body is the sum of the kinetic energies of each
component of the rigid body. Thus
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Physics 235
Chapter 11
T
(
)
"1
% 1
= ( # m! v! 2 & = ( m! {V + ) * r! } • {V + ) * r! } =
' 2 !
! $2
1
= ( m! V 2 + 2V • {) * r! } + {) * r! } • {) * r! }
2 !
(
)
Let us now examine the three terms in this expression:
1
1
1
m! V 2 = V 2 " m! = MV 2
"
2 !
2
2
!
%
(
1
m! ( 2V • {" # r! }) = V • &" # $ ( m! r! ) ) = V • " # ( MR ) = 0
$
2 !
!
'
*
{
(
}
)
1
1
2
m! ({" # r! } • {" # r! }) = $ m! " 2 r! 2 % {" • r! } =
$
2 !
2 !
=
&
)
1
m! ( $ " i 2 $ r! ,k 2 % $ (" i r! ,i )$ " j r! , j + =
$
2 !
' i
*
k
i
j
=
&
)1
1 1
2
.$ m! ( , ij $ r! ,k % r! ,i r! , j + 2 " i" j = $ I ij " i" j
$
2 i, j 0/ !
2 i, j
'
* 03
k
(
)
The second term is zero, if we choose the origin of the rotating coordinate system to coincide
with the center of mass of the rigid object.
Using the previous expressions, we can now rewrite the total kinetic energy of the rigid
object as
T=
1
1
MV 2 + " I ij ! i! j = TCM + Trot
2
2 i, j
The quantity Iij is called the inertia tensor, and is a 3 x 3 matrix:
$
2
2
& " m! r! ,2 + r! ,3
& !
&
{I } = & # " m! r! ,2 r! ,1
!
&
&
& # " m! r! ,3 r! ,1
%
!
(
)
# " m! r! ,1 r! ,2
# " m! r! ,1 r! ,3
!
" m! ( r!
!
2
,1
+ r! ,32
# " m! r! ,3 r! ,2
!
- 2 -
!
)
# " m! r! ,2 r! ,3
!
" m! ( r!
!
2
,1
+ r! ,2 2
'
)
)
)
)
)
)
)
(
)
Physics 235
Chapter 11
Based on the definition of the inertia tensor we make the following observations:
• The tensor is symmetric: Iij = Iji. Of the 9 parameters, only 6 are free parameters.
• The non-diagonal tensor elements are called products of inertia.
• The diagonal tensor elements are the moments of inertia with respect to the three
coordinate axes of the rotating frame.
Angular Momentum
The total angular momentum L of the rotating rigid object is equal to the vector sum of the
angular momenta of each component of the rigid object. The ith component of L is equal to
Li
(
)
= # ( r! " p! )i = # ( r! " m! ($ " r! ))i = # m! ( r! " $ " r! )i = # m! r! 2$ % r! ( r! • $ ) i =
!
!
!
!
*(
(&
= # m! 'r! 2$ i % r! ,i # r! , j $ j + = # $ j # m! r! 2- ij % r! ,i r! , j = # I ij $ j
(, j
!
j
!
j
)(
(
)
{
}
This equation clearly shows that the angular momentum is in general not parallel to the angular
velocity. An example of a system where the angular momentum is directed in a different from
the angular velocity is shown in Figure 1.
Figure 1. A rotating dumbbell is an example of a system in which the angular velocity is not
parallel to the angular momentum.
The rotational kinetic energy can also be rewritten in terms of the angular momentum:
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Physics 235
Chapter 11
Trot =
#
& 1
1
1
1
I ij ! i! j = " ! i % " I ij ! j ( = " ! i Li = (! • L )
"
2 i, j
2 i
2
$ j
' 2 i
Principal Axes
We always have the freedom to choose our coordinate axes such that the problem we are
trying solve is simplified. When we are working on problems that involve the use of the inertia
tensor, we can obtain a significant simplification if we can choose our coordinate axes such that
the non-diagonal elements are 0. In this case, the inertia tensor would be equal to
! I1
{I } = ## 0
"0
0
I2
0
0$
0&
&
I3 %
For this inertia tensor we get the following relation between the angular momentum and the
angular velocity:
Li = I i! i
The rotational kinetic energy is equal to
Trot =
1
I i! i 2
"
2 i
The axes for which the non-diagonal matrix elements vanish are called the principal axes of
inertia.
The biggest problem we are facing is how do we determine the proper coordinate axes? If
the angular velocity vector is directed along one of the three coordinate axes that would get rid of
the non-diagonal inertia tensor elements, we expect to see the following relation between the
angular velocity vector and the angular momentum:
L = I!
Substituting the general form of the inertia tensor into this expression, we must require that
L1 = I ! 1 = I11! 1 + I12! 2 + I13! 3
L2 = I ! 2 = I 21! 1 + I 22! 2 + I 23! 3
L3 = I ! 3 = I 31! 1 + I 32! 2 + I 33! 3
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Physics 235
Chapter 11
This set of equations can be rewritten as
( I11 ! I )"1 + I12" 2 + I13" 3 = 0
I 21" 1 + ( I 22 ! I )" 2 + I 23" 3 = 0
I 31" 1 + I 32" 2 + ( I 33 ! I )" 3 = 0
This set of equations only has a non-trivial solution if the determinant of the coefficients vanish.
This requires that
I11 ! I
I 21
I 31
I12
I 22 ! I
I 32
I13
I 23 = 0
I 33 ! I
This requirement leads to three possible values of I. Each of these corresponds to the moment of
inertia about one of the principal aces.
Example: Problem 11.13
A three-particle system consists of masses mi and coordinates (x1, x2, x3) as follows:
(b, 0,b )
(b,b, !b )
( !b,b, 0 )
m1 = 3m
m2 = 4m
m3 = 2m
Find the inertia tensor, the principal axes, and the principal moments of inertia.
We get the elements of the inertia tensor from Eq. 11.13a:
(
)
( )
( )
I11 = " m! x!2 ,2 + x!2 ,3
!
( )
= 3m b 2 + 4m 2b 2 + 2m b 2 = 13mb 2
Likewise I 22 = 16mb2 and I 33 = 15mb2
I12 = I 21 = ! # m" x" ,1 x" ,2
"
( )
( )
= !4m b 2 ! 2m !b 2 = !2mb 2
Likewise I13 = I 31 = mb2
and I 23 = I 32 = 4mb2
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Physics 235
Chapter 11
Thus the inertia tensor is
" 13 !2 1 %
{I } = mb $$ !2 16 4 ''
$# 1
4 15 '&
2
The principal moments of inertia are gotten by solving
!2
1 &
#13 ! "
%
mb % !2
16 ! "
4 (( = 0
%$ 1
4
15 ! " ('
2
Expanding the determinant gives a cubic equation in λ:
! 3 " 44 ! 2 + 622! " 2820 = 0
Solving numerically gives
!1 = 10.00
!2 = 14.35
!3 = 19.65
Thus the principal moments of inertia are I1 = 10 mb 2
I 2 = 14.35 mb 2
I 3 = 19.65 mb 2
To find the principal axes, we substitute into (see example 11.3):
(13 ! "i ) # 1i ! 2# 2i + # 3i = 0
!2 # 1i + (16 ! "i ) # 2i + 4# 3i = 0
# 1i + 4# 2i + (15 ! "i ) # 3i = 0
For i = 1, we have ( !1 = 10 )
3! 11 " 2! 21 + ! 31 = 0
"2! 11 + 6! 21 + 4! 31 = 0
! 11 + 4! 21 + 5! 31 = 0
Solving the first for ! 31 and substituting into the second gives
! 11 = ! 21
Substituting into the third now gives
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Physics 235
Chapter 11
! 31 = "! 21
or
! 11 : ! 21 : ! 31 = 1 : 1 : " 1
So, the principal axis associated with I1 is
1
3
( x̂ + ŷ ! ẑ )
Proceeding in the same way gives the other two principal axes:
i = 2 : !.81x̂ + .29 ŷ ! .52 ẑ
i = 3 : !.14 x̂ + .77 ŷ + .63 ẑ
We note that the principal axes are mutually orthogonal, as they must be.
Our observation in problem 11.13 that the principal vectors are orthogonal is true in general.
We can prove this in the following manner. For the mth principal moment the following relations
must hold:
Lim = I m! im
Lim = " I ik ! km
k
Combining these two equations we obtain
"I
ik
! km = I m! im
k
Now multiply both sides of this equation by ωin and sum over i:
"I
ik
i,k
! km! in = " I m! im! in
i
A similar relation can be obtained for the nth principal moment, multiplied by ωkm and summed
over k:
"I
i,k
ki
! in! km = " I n! km! kn
k
If we subtract the last equation from the one-before-last equation we obtain the following result:
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Physics 235
Chapter 11
"I
i,k
ik
! km! in # " I ki! in! km = " ( I ik # I ki )! in! km = 0 =
i,k
i,k
= " I m! im! in # " I n! im! in = ( I m # I n ) " ! im! in
i
i
i
Assuming that the principal momenta are distinct, the previous equation can only be correct if
"!
im
! in = ! m • ! n = 0
i
which shows the principal axes are orthogonal.
Transformations of the Inertia Tensor
In our discussion so far we have assumed that the origin of the rotating reference frame
coincidence with the center of mass of the rigid object. In this Section we will examine what
will change if we do not make this assumption.
Consider the two coordinate systems shown in Figure 2. One reference frame, the x frame,
has its origin O coincide with the center of mass of the rigid object; the second reference frame,
the X frame, has an origin Q that is displaced with respect to the center of mass of the rigid
object.
Figure 2. Two coordinate systems used to describe our rigid body.
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Physics 235
Chapter 11
The inertia tensor Jij in reference frame X is defined in the same way as it was defined
previously:
%
(
J ij = # m! ' " ij # X! ,k 2 $ X! ,i X! , j *
&
)
!
k
The coordinates in the X frame are related to the coordinates in the x frame in the following way:
Xi = ai + xi
Using this relation we can express the inertia tensor in reference frame X in terms of the
coordinates in reference frame x:
J ij
%
(
2
= # m! ' " ij # ( ak + x! ,k ) $ ( ai + x! ,i ) a j + x! , j * =
&
)
!
k
(
)
%
(
%
(
= # m! ' " ij # x! ,k 2 $ x! ,i x! , j * + # m! ' " ij # ak 2 + 2ak x! ,k $ ai a j + ai x! , j + a j x! ,i * =
&
) !
&
)
!
k
k
) (
(
)
%
(
%
(
= I ij + # m! ' " ij # ak 2 $ ai a j * + # m! ' 2" ij # ( 2ak x! ,k ) $ ai x! , j $ a j x! ,i *
&
) !
&
)
!
k
k
The last term on the right-hand side is equal to 0 since the origin of the coordinate system x
coincides with the center of mass of the object:
" m! x!
!
,k
= Mrcm = 0
The relation between the inertia tensor in reference frame X and the inertia tensor in reference
frame x is thus given by
%
(
J ij = I ij + # m! ' " ij # ak 2 $ ai a j * = I ij + M " ij a 2 $ ai a j
&
)
!
k
(
)
This relation is called the Steiner's parallel-axis theorem and is one example of how coordinate
transformations affect the inertia tensor.
The transformation discussed so far is a simple translation. Other important transformations
are rotations. In Chapter 1 we discussed many examples of rotations, and determined that the
most general way to express rotations is by using the rotation matrix λ:
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Physics 235
Chapter 11
xi = " ! ji x ' j
j
Since this transformation rule is valid for vectors in general, the same rule can be used to
describe the transformation of the angular momentum and angular velocity vectors:
Li = " ! ji L ' j
j
! i = # " ji! ' j
j
In order to determine the relation between the inertia tensor in the two coordinate frames, we use
the fact that the angular momentum is the product of the inertia tensor and the angular velocity,
in both frames:
Lk = " I kl ! l
l
and
L 'k = " I 'kl ! 'l
l
In order to relate the inertia tensors, we use the coordinate transformations for L and ω:
"!
j
jk
L ' j = " I kl " !ml # 'm
l
m
This equation can be simplified if we multiply each side by λik and sum over k:
#
" ! %$ " !
ik
k
j
&
L
'
jk
j(
'
)
,
= " *" ! jk !ik L ' j - = " / ji L ' j = L 'i =
j + k
. j
(
)
{
}
)
,
#
&
= " !ik % " I kl " !ml 0 'm ( = " *" !ik !ml I kl - 0 'm
$ l
' m + k,l
k
m
.
where we have used the orthogonal properties of the rotation matrix. Using the relation between
the angular momentum and the angular velocity in the rotated coordinate frame we see that the
inertia tensors in the two coordinate frames are related as follows:
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Physics 235
Chapter 11
I 'im = " !ik !ml I kl = " !ik I kl ! t lm
k,l
k,l
where λt is the transposed matrix. In tensor notation we can rewrite this relation as
{I '} = {!} {I } {! t }
It turns out that for any inertia tensor we can find a rotation such that the inertia tensor in the
rotated frame is a diagonal matrix (all non-diagonal elements are equal to 0).
We thus have seen two different approaches to diagonalize the inertia tensor: 1) find the
principal axes of inertia, and 2) find the proper rotation matrix.
Example: Problem 11.16
Consider the following inertia tensor:
1
"1
&
$ 2 ( A + B ) 2 ( A ! B)) 0 $
$
$
1
$1
{I } = # ( A ! B)) ( A + B ) 0 $'
2
$2
$
0
0
C$
$
$%
$(
Perform a rotation of the coordinate system by an angle θ about the x3 axis. Evaluate the
transformed tensor elements, and show that the choice θ = π/4 renders the inertia tensor diagonal
with elements A, B, and C.
The rotation matrix is
$ cos "
( ! ) = && # sin "
&% 0
sin " 0 '
cos " 0 ))
0
1 )(
(1)
The moment of inertia tensor transforms according to
(" ! ) = ( # ) (" ) ( # t )
That is
- 11 -
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Physics 235
Chapter 11
$ cos "
( I ! ) = && # sin "
&% 0
1
$1
'
& 2 ( A + B) 2 ( A # B) 0 )
sin " 0 ' &
) $cos "
1
1
)
&
cos " 0 ) ( A # B )
A + B ) 0 ) && sin "
(
&2
)
2
&
0
1 )( &
0
0
C )) % 0
&
&%
)(
# cos !
= %% " sin !
%$ 0
1
#1
% 2 ( A + B ) cos ! + 2 ( A " B ) sin !
sin ! 0 & %
1
1
cos ! 0 (( % ( A " B ) cos ! + ( A + B ) sin !
%2
2
0
1 (' %
0
%
%$
# sin " 0 '
cos " 0 ))
0
1 )(
1
1
A + B ) sin ! + ( A " B ) cos !
(
2
2
1
1
" ( A " B ) sin ! + ( A + B ) cos !
2
2
0
"
&
0(
(
0(
(
C ((
('
1
#1
2
2
% 2 ( A + B ) cos ! + ( A " B ) cos ! sin ! + 2 ( A + B ) sin !
%
1
1
=%
" ( A " B ) sin 2 ! + ( A " B ) cos 2 !
%
2
2
%
0
%
$%
1
1
A " B ) cos 2 ! " ( A " B ) sin 2 !
(
2
2
1
1
2
A + B ) sin ! " ( A " B ) sin ! cos ! + ( A + B ) cos 2 !
(
2
2
0
&
0(
(
0(
(
C ((
('
or
$ 1
& 2 ( A + B ) + ( A " B ) cos # sin #
&
(I ! ) = && " 12 ( A " B) sin 2 # + 12 ( A " B) cos2 #
&
0
&
&%
If ! = " 4 , sin ! = cos ! = 1
1
1
A " B ) cos 2 # " ( A " B ) sin 2 #
(
2
2
1
( A + B) " ( A " B) cos # sin #
2
0
'
0)
)
0)
)
C ))
)(
(3)
2 . Then,
"A 0 0 %
(I ! ) = $$ 0 B 0 ''
$# 0 0 C '&
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Physics 235
Chapter 11
Euler Angles
Any rotation between different coordinate systems can be expressed in terms of three
successive rotations around the coordinate axes. When we consider the transformation from the
fixed coordinate system x' to the body coordinate system x, we call the three angles the Euler
angles φ, θ, and ψ (see Figure 3).
Figure 3. The Euler angles used to transform the fixed coordinate system x' into the body
coordinate system x.
The total transformation matrix is the product of the individual transformations (note order)
$ cos "
! = & # sin "
&
% 0
sin "
0' $ 1
0
)
&
0 0 cos *
)&
1 ( % 0 sin *
0 ' $ cos + sin + 0 '
sin * ) & # sin + cos + 0 ) =
)&
)
cos * ( % 0
0
1(
$ cos " cos + # cos * sin + sin "
= & # sin " cos + # cos * sin + cos "
&
sin * sin +
%
cos " sin + + cos * cos + sin "
cos "
0
# sin " sin + + cos * cos + cos "
# sin * cos +
sin " sin * '
cos " sin * )
)
cos * (
With each of the three rotations we can associate an angular velocity ω. To express the angular
velocity in the body coordinate system, we can use Figure 3c.
•
ωφ: Figure 3c shows that the angular velocity ωφ is directed in the x2''' - x3''' plane. Its
projection along the x3''' axis, which is also the x3 axis, is equal to
!!3 = !! cos "
The projection along the x2''' axis is equal to
!!2 ''' = !! sin "
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Physics 235
Chapter 11
Figure 3c shows that when we project this projection along the x1 and x2 axes we obtain the
following components in the body coordinate system:
!!1 = !!2 ''' sin " = !! sin # sin "
!!2 = !!2 ''' cos " = !! sin # cos "
•
ωθ: Figure 3c shows that the angular velocity ωθ is directed in the x1''' - x2''' plane. Its
projection along the x3''' axis, which is also the x3 axis, is equal to 0.
!!3 = 0
Figure 3c shows that when we project ωθ along the x1 and x2 axes we obtain the following
components in the body coordinate system:
!!1 = !! cos "
!!2 = "!! sin #
•
ωψ: Figure 3c shows that the angular velocity ωψ is directed along the x3''' axis, which is also
the x3 axis. The components along the other body axes are 0. Thus:
!! 1 = 0
!! 2 = 0
!! 3 = !!
The angular velocity, in the body frame, is thus equal to
" ! 1 % " (!1 + )!1 + *! 1 % " (! sin ) sin * + )! cos * %
! = $ ! 2 ' = $ (!2 + )!2 + *! 2 ' = $ (! sin ) cos * + )! sin * '
' $
'
$ ' $
'&
(! cos ) + *!
# ! 3 & $# (!3 + )!3 + *! 3 '& $#
The Force-Free Euler Equations
Let's assume for the moment that the coordinate axis correspond to the principal axes of the
body. In that case, we can write the kinetic energy of the body in the following manner:
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Physics 235
Chapter 11
1
I i! i 2
"
2 i
T=
where Ii are the principal moments of the rigid body. If for now we consider that the rigid object
is carrying out a force-free motion (U = 0) then the Lagrangian L will be equal to the kinetic
energy T. The motion of the object can be described in terms of the Euler angles, which can
serve as the generalized coordinates of the motion. Consider the three equations of motion for
the three generalized coordinates:
•
The Euler angle φ: Lagrange's equation for the coordinate φ is
0=
!$ i d
!$ i
!L d !L !T d !T
!T !$ i d
!T !$ i
#
=
#
=%
# %
= % I i$ i
# % I i$ i
=0
!" dt !"! !" dt !"!
dt i !$ i !"!
!" dt i
!"!
i !$ i !"
i
Differentiating the angular velocity with respect to the coordinate φ we find
!" 1
=0
!#
!" 2
=0
!#
!" 3
=0
!#
!" 1
=0
!#!
!" 2
=0
!#!
!" 3
= cos $
!#!
and Lagrange's equation becomes
d
{ I 3! 3 cos" } = 0
dt
•
The Euler angle θ: Lagrange's equation for the coordinate θ is
0=
!$ i d
!$ i
!T d !T
!T !$ i d
!T !$ i
#
=%
# %
= % I i$ i
# % I i$ i
=0
!" dt !"!
dt i !$ i !"!
!" dt i
!"!
i !$ i !"
i
Differentiating the angular velocity with respect to the coordinate θ we find
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Chapter 11
!" 1 !
= $ cos # sin %
!#
!" 2 !
= $ cos # cos %
!#
!" 3
= &$! sin #
!#
!" 1
= cos %
!#!
!" 2
= & sin %
!#!
!" 3
=0
!#!
and Lagrange's equation becomes
d
!! ({ I1" 1 sin # + I 2" 2 cos # } cos $ % I 3" 3 sin $ ) % { I1" 1 cos # % I 2" 2 sin # } = 0
dt
•
The Euler angle ψ: Lagrange's equation for the coordinate ψ is
0=
!$ i d
!$ i
!T d !T
!T !$ i d
!T !$ i
#
=%
# %
= % I i$ i
# % I i$ i
=0
!" dt !"!
dt i !$ i !"!
!" dt i
!"!
i !$ i !"
i
Differentiating the angular velocity with respect to the coordinate ψ we find
!" 1 !
= $ sin % cos # & %! sin # = " 2
!#
!" 2
= &$! sin % sin # & %! cos # = &" 1
!#
!" 3
=0
!#
!" 1
=0
!%!
!" 2
=0
!%!
!" 3
=1
!%!
and Lagrange's equation becomes
I1! 1! 2 " I 2! 2! 1 "
d
d
I 3! 3 } = ( I1 " I 2 )! 1! 2 " { I 3! 3 } = ( I1 " I 2 )! 1! 2 " I 3!! 3 = 0
{
dt
dt
Of all three equations of motion, the last one is the only one to contain just the components of the
angular velocity. Since our choice of the x3 axis was arbitrary, we expect that similar relations
should exist for the other two axes. The set of three equation we obtain in this way are called the
Euler equations:
( I1 ! I 2 )"1" 2 ! I 3"! 3 = 0
( I 2 ! I 3 )" 2" 3 ! I1"! 1 = 0
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Chapter 11
( I 3 ! I1 )" 3"1 ! I 2"! 2 = 0
As an example of how we use Euler's equations, consider a symmetric top. The top will have
two different principal moments: I1 = I2 and I3. In this case, the first Euler equations reduces to
I 3!! 3 = 0
or
! 3 ( t ) = constant = ! 3
The other two Euler equations can be rewritten as
#I "I
&
!! 1 = " % 3 1 ! 3 ( ! 2 = ")! 2
$ I1
'
#I "I
&
!! 2 = % 3 1 ! 3 ( ! 1 = )! 1
$ I1
'
This set of equations has the following solution:
! 1 ( t ) = A cos"t
! 2 ( t ) = A sin "t
The magnitude of the angular velocity of the system is constant since
! =
(!
2
1
( t ) + ! 2 2 ( t ) + ! 32 ) =
" 2 + ! 32
The angular velocity vector traces out a cone in the body frame (it precesses around the x3 axis see Figure 4). The rate with which the angular velocity vector precesses around the x3 axis is
determined by the value of Ω. When the principal moment I3 and the principal moment I1 are
similar, Ω will become very small.
Since we have assumed that there are no external forces and torques acting on the system, the
angular momentum of the system will be constant in the fixed reference frame. If the angular
momentum is initially pointing along the x'3 axis it will continue to point along this axis (see
Figure 5). Since there are no external forces and torques acting on the system, the rotation
kinetic energy of the system must be constant. Thus
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Physics 235
Chapter 11
1
Trot = ! • L
2
Since the angle between the angular velocity vector and the angular momentum vector must be
constant, the angular velocity vector must trace out a space cone around the x'3 (see Figure 5).
Figure 4. The angular velocity of a force-free Figure 5. The angular velocity of a force-free
symmetric top, precessing around the x3 axis in symmetric top, tracing out a space-cone around
the body frame.
the x'3 axis in the body frame.
Example: Problem 11.27
A symmetric body moves without the influence of forces or torques. Let x3 be the symmetry
axis of the body and L be along x3'. The angle between the angular velocity vector and x3 is α.
Let ω and L initially be in the x2-x3 plane. What is the angular velocity of the symmetry axis
about L in terms of I1, I3, ω, and α?
Initially:
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Physics 235
Chapter 11
L1 = 0 = I1! 1
L2 = L sin " = I1! 2 = I1! sin #
L3 = L xos " = I 3! 3 = I 3! cos #
Thus
tan ! =
L2 I1
= tan "
L3 I 3
(1)
From Eq. (11.102)
! 3 = "! cos # + $!
Since ! 3 = ! cos " , we have
!! cos " = # cos $ % &!
(2)
From Eq. (11.131)
!! = "# = "
I 3 " I1
$3
I1
(2) becomes
I
!! cos " = 3 # cos $
I1
(3)
From (1), we may construct the following triangle
from which cos ! =
I3
12
#$ I 32 + I12 tan 2 " %&
Substituting into (3) gives
"
!! =
I1
I12 sin 2 # + I 32 cos 2 #
The Euler Equations in a Force Field
When the external forces and torques acting on the system are not equal to 0, we can not use
the method we have used in the previous section to obtain expressions for the angular velocity
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Physics 235
Chapter 11
and acceleration. The procedure used in the previous section relied on the fact that the potential
energy U is 0 in a force-free environment, and therefore, the Lagrangian L is equal to the kinetic
energy T.
When the external forces and torques are not equal to 0, the angular momentum of the system
is not conserved:
! dL $
=N
#" dt &%
fixed
Note that this relation only holds in the fixed reference frame since this is the only good inertial
reference frame. In Chapter 10 we looked at the relation between parameters specified in the
fixed reference frame compared to parameters specified in the rotating reference frame, and we
can use this relation to correlate the rate of change of the angular momentum vector in the fixed
reference frame with the rate of change of the angular momentum vector in the rotating reference
frame:
! dL $
! dL $
N =# &
=# &
+' ( L
" dt % fixed " dt % rotating
This relation can be used to generate three separate relations by projecting the vectors along the
three body axes:
N1 =
dL1
dL
+ (! " L )1 = 1 + (! 2 L3 # ! 3 L2 ) = I1!! 1 # ( I 2 # I 3 )! 2! 3
dt
dt
N2 =
dL2
dL
+ (! " L )2 = 2 + (! 3 L1 # ! 1 L3 ) = I 2!! 2 # ( I 3 # I1 )! 3! 1
dt
dt
N3 =
dL3
dL
+ (! " L )3 = 3 + (! 1 L2 # ! 2 L1 ) = I 3!! 3 # ( I1 # I 2 )! 1! 2
dt
dt
These equations are the Euler equations for the motion of the rigid body in a force field. In the
absence of a torque, these equations reduce to the force-free Euler equations.
Example: Motion of a Symmetric Top with One Point Fixed
In order to describe the motion of a top, which has its tip fixed, we use two coordinate
systems whose origins coincide (see Figure 6). Since the origins coincide, the transformation
between coordinate systems can be described in terms of the Euler angles, and the equations of
motion will be the Euler equations:
- 20 -
Physics 235
Chapter 11
( I1 ! I 2 )"1" 2 ! I 3"! 3 = 0
( I 2 ! I 3 )" 2" 3 ! I1"! 1 = 0
( I 3 ! I1 )" 3"1 ! I 2"! 2 = 0
Figure 6. Spinning top with fixed tip.
Since the top is symmetric around the x3 axis, it principal moments of inertia with respect to the
x1 and x2 axes are identical. The Euler equations now become
I 3!! 3 = 0
( I1 ! I 3 )" 2" 3 ! I1"! 1 = 0
! ( I1 ! I 3 )" 3" 1 ! I1"! 2 = 0
The first equation immediately tells us that
! 3 = constant
The motion of the top is often described in terms of the motion of its rotating axes. The kinetic
energy of the system is equal to
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Physics 235
T=
Chapter 11
(
)
(
)
2
1
1
1
1
1
I i! i 2 = I1 ! 12 + ! 2 2 + I 3! 32 = I1 #! 2 sin 2 $ + $! 2 + I 3 (#! cos $ + %! )
"
2 i
2
2
2
2
The potential energy of the system, assuming the center of mass of the top is located a distance h
from the tip, is equal to
U = Mgh cos !
The Lagrangian is thus equal to
L = T !U =
(
)
2
1
1
I1 "! 2 sin 2 # + #! 2 + I 3 ("! sin # + $! ) ! Mgh cos #
2
2
The Lagrangian does not depend on φ and ψ, and thus
!L
d !L
=0=
!"
dt !"!
!L
!L
=0=
!"
!"!
We thus conclude that the angular momenta associated with the Euler angles φ and ψ are
constant:
p! =
p! =
"L
= constant
"!!
"L
= I 3 (!! + #! cos $ ) = I 3% 3 =constant
"!!
Expressing the momenta in terms of the Euler angles φ and ψ allows us to express the rate of
change of these Euler angles in terms of the angular momenta:
!! =
!! =
(p
"
p! " p# cos $
I1 sin 2 $
)
# p! cos $ cos $
I1 sin $
2
= "! cos $
Since there are no non-conservative forces acting on the top, the total energy E of the system is
conserved. Thus
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Physics 235
Chapter 11
E = T +U =
(
)
2
1
1
I1 !! 2 sin 2 " + "! 2 + I 3 (!! cos " + #! ) + Mgh cos " = constant
2
2
The total energy can be rewritten in terms of the angular momenta:
E =
(
)
(
)
2
1
1
I1 !! 2 sin 2 " + "! 2 + I 3 (!! cos " + #! ) + Mgh cos " =
2
2
=
1
1
I1 !! 2 sin 2 " + "! 2 + I 3$ 32 + Mgh cos " =
2
2
=
1 !2 1 !2 2
1
I1" + I1! sin " + I 3$ 32 + Mgh cos " =
2
2
2
(
1
1 p! % p# cos "
= I1"! 2 +
2
2
I1 sin 2 "
)
2
+
1
I 3$ 32 + Mgh cos "
2
Since the angular velocity with respect to the x3 axis is constant, we can subtract it from the
energy E to get the effective energy E' (note: this is equivalent to choosing the zero point of the
energy scale). Thus
(
1
1
1 p$ ! p% cos #
E ' = E ! I 3" 32 = I1#! 2 +
2
2
2
I1 sin 2 #
)
2
+ Mgh cos # = constant
The effective energy only depends on the angle θ and on dθ/dt since the angular momenta are
constants. The manipulations we have carried out have reduced the three-dimensional problem
to a one-dimensional problem. The first term in the effective energy is the kinetic energy
associated with the rotation around the x1 axis. The last two terms depend only on the angle θ
and not on the angular velocity dθ/dt. These terms are what we could call the effective potential
energy, defined as
(
1 p" # p$ cos !
V (! ) =
2
I1 sin 2 !
)
2
+ Mgh cos !
The effective potential becomes large when the angle approaches 0 and π. The angular
dependence of the effective potential is shown in Figure 7. If the total effective energy of the
system is E1', we expect the angle θ to vary between θ1 and θ2. We thus expect that the angle of
inclination of the top will vary between these two extremes.
- 23 -
Physics 235
Chapter 11
Figure 7. The effective potential of a rotating top.
The minimum effective energy that the system can have is E2'. The corresponding angle can
be found by requiring
!V
!"
=0
" =" 0
This requirement can be rewritten as a quadratic equation of a parameter β, where β is defined as
! = p" # p$ cos % 0
In general, there are two solutions to this quadratic equation. Since β is a real number, the
solution must be real, and this requires that
1!
4MghI1 cos " 0
$0
p# 2
This equation can be rewritten as
p! 2 = ( I 3" 3 ) # 4MghI1 cos $ 0
2
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Physics 235
Chapter 11
When we study a spinning top, the spin axis is oriented such that θ0 < π/2. The previous
equation can then be rewritten as
I 3! 3 " 4MghI1 cos # 0
or
!3 "
2
I3
MghI1 cos # 0
There is thus a minimum angular velocity the system must have in order to produce stable
precession. The rate of precession can be found by calculating
!! =
p! " p# cos $ 0
I1 sin $ 0
2
=
%
I1 sin 2 $ 0
Since β has two possible values, we expect to see two different precession rates: one resulting in
fast precession, and one resulting in slow precession.
When the angle of inclination is not equal to θ0, the system will oscillate between two
limiting values of θ. The precession rate will be a function of θ and can vary between positive
and negative values, depending on the values of the angular momenta. The phenomenon is
called nutation, and possible nutation patterns are shown in Figure 8. The type of nutation
depends on the initial conditions.
Figure 8. The nutation of a rotating top.
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Physics 235
Chapter 11
Example: Problem 11.30
Investigate the equation for the turning points of the nutational motion by setting dθ/dt = 0 in
the equation of the effective energy. Show that the resulting equation is cubic in cosθ and has
two real roots and one imaginary root.
If we set !! = 0 in the equation for the effective energy we obtain
( P $ P cos " ) + Mgh cos "
E ! = V (" ) =
2 I (1 $ cos " )
2
#
%
2
(1)
12
Re-arranging, this equation can be written as
( 2Mgh I12 ) cos3 ! " ( 2E# I12 + P$2 ) cos2 ! + 2 ( P% P$ " Mgh I12 ) cos ! + ( 2E# I12 " P%2 ) = 0
(2)
which is cubic in cosθ.
V(θ) has the form shown in the diagram. Two of the roots occur in the region !1 " cos # " 1 , and
one root lies outside this range and is therefore imaginary.
Stability of Rigid-Body Rotations
The rotation of a rigid body is stable if the system, when perturbed from its equilibrium
condition, carries out small oscillations about it. Consider we use the principal axes of rotation
to describe the motion, and we choose these axes such that I3 > I2 > I1. If the system rotates
around the x1 axis we can write the angular velocity vector as
! = ! 1 x̂1
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Physics 235
Chapter 11
Consider what happens when we apply a small perturbation around the other two principal axes
such that the angular velocity becomes
! = ! 1 x̂1 + " x̂2 + µ x̂3
The corresponding Euler equations are
( I1 ! I 2 )"1" 2 ! I 3"! 3 = ( I1 ! I 2 ) #"1 ! I 3 µ! = 0
( I 2 ! I 3 )" 2" 3 ! I1"! 1 = ( I 2 ! I 3 ) #µ ! I1"! 1 = 0
( I 3 ! I1 )" 3"1 ! I 2"! 2 = ( I 3 ! I1 ) µ"1 ! I 2 #! = 0
Since we are talking about small perturbations from the equilibrium state, λµ will be small and
can be set to 0. The second equation can thus be used to conclude that
! 1 = constant
The remaining equations can be rewritten as
#I !I
&
µ! = % 1 2 " 1 ( )
$ I3
'
$I "I
'
!! = & 3 1 # 1 ) µ
% I2
(
The last equation can be differentiated to obtain
$ ( I " I )( I " I )
'
d !! $ I 3 " I1 ' d µ $ I 3 " I1 ' $ I1 " I 2 '
!!! =
=&
#1 )
=&
#1 ) &
# 1 ) ! = " & 3 1 2 1 # 12 ) !
dt % I 2
I2 I3
( dt % I 2
( % I3
(
%
(
The term within the parenthesis is positive since we assumed that I3 > I2 > I1. This differential
equation has the following solution:
! ( t ) = A! cos ( "1t ) + B! sin ( "1t )
where
- 27 -
Physics 235
Chapter 11
( I 3 # I1 ) ( I 2 # I1 )
!1 = " 1
I2 I3
When we look at the perturbation around the x3 axis we find the following differential equation
µ!! =
# ( I ! I )( I ! I )
&
d µ! # I1 ! I 2 & d ) # I1 ! I 2 & # I 3 ! I1 &
=%
"1 (
=%
"1 ( %
" 1 ( µ = ! % 2 1 3 1 " 12 ( µ
dt $ I 3
I2 I3
' dt $ I 3
' $ I2
'
$
'
The solution of the second-order differential equation is
µ ( t ) = Aµ cos ( !1t ) + Bµ sin ( !1t )
We see that the perturbations around the x2 axis and the x3 axis oscillate around the equilibrium
values of λ = µ = 0. We thus conclude that the rotation around the x1 axis is stable.
Similar calculations can be done for rotations around the x2 axis and the x3 axis. The
perturbation frequencies obtained in those cases are equal to
( I1 # I 2 ) ( I 3 # I 2 )
!2 = " 2
I 3 I1
( I 2 # I 3 ) ( I1 # I 3 )
!3 = " 3
I1 I 2
We see that the first frequency is an imaginary number while the second frequency is a real
number. Thus, the rotation around the x3 axis is stable, but the rotation around the x2 axis is
unstable.
Example: Problem 11.34
Consider a symmetrical rigid body rotating freely about its center of mass. A frictional
torque (Nf = -bω) acts to slow down the rotation. Find the component of the angular velocity
along the symmetry axis as a function of time.
The Euler equation, which describes the rotation of an object about its symmetry axis, say
the x axis, is
(
)
I x !! x " I y " I z ! y! z = N x
- 28 -
Physics 235
Chapter 11
where N x = !b " x is the component of torque along Ox. Because the object is symmetric about
the x axis, we have I y = I z , and the above equation becomes
d! x
Ix
= "b ! x
dt
# !x = e
- 29 -
"
b
t
Ix
! x0